Conjectured upper bound for C2: C2 ≤ 2√(2/3)

Prove that C_2 ≤ 2√(2/3), where C_2 = c(u_u, u_0) is the universal commuting dilation constant for pairs of contractions, i.e., show that every pair of contractions dilates to a pair of commuting normal operators whose norms are at most 2√(2/3).

Background

Using known exact values c(u_u, u_f) = d/√(2d−1) for d=2 and inequalities linking c(u_f, u_0) to limits of c(U^{(N)}, u_0) for random Haar unitary pairs, the authors derive semi-rigorously that C_2 ≤ (2/√3)·c(u_f, u_0). Their experiments suggest c(u_f, u_0) = √2, which would imply C_2 ≤ 2√(2/3) < 2.

A proof of this conjectured bound would strictly improve the best known upper bound C_2 ≤ 2 and would align with empirical observations obtained via semidefinite programming computations of dilation constants for random unitary pairs.